Fluid Mechanics

JEE Main level, single correct, medium

147 of 192

Question 147 of 192

A capillary tube is immersed vertically in water and the height of the water column is x⁡x⁡. When this arrangement is taken into a mine of depth d, the height of the water column is y⁡y⁡. If R is the radius of earth, the ratio x⁡y⁡\dfrac{x⁡}{y⁡} is :

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Answer: (D) (1−dR)\left(1-\dfrac{\text{d}}{\text{R}}\right)

We know that

h=2T cosθρgr\text{h}=\dfrac{2\text{T cos}\theta }{\rho \text{gr}}

∴x=2Tcosθρgr∴\text{x}=\dfrac{2\text{T}cos\theta }{\rho \text{gr}}

Now, value of geff at depth 'd' is

geff=g(1−dR){\text{g}}_{\text{eff}}=\text{g}\left(1-\dfrac{\text{d}}{\text{R}}\right)

∴y=2Tcosθρgeff⋅r∴\text{y}=\dfrac{2\text{T}cos\theta }{\rho {\text{g}}_{\text{eff}}\cdot \text{r}}

∴xy=geffg=1−dR∴\dfrac{\text{x}}{\text{y}}=\dfrac{{\text{g}}_{\text{eff}}}{\text{g}}=1-\dfrac{\text{d}}{\text{R}}

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